We construct models of universe with a generalized equation of state p = ( \alpha \rho + k \rho ^ { 1 + 1 / n } ) c ^ { 2 } having a linear component and a polytropic component .
The linear equation of state p = \alpha \rho c ^ { 2 } describes radiation ( \alpha = 1 / 3 ) , pressureless matter ( \alpha = 0 ) , stiff matter ( \alpha = 1 ) , and vacuum energy ( \alpha = -1 ) .
The polytropic equation of state p = k \rho ^ { 1 + 1 / n } c ^ { 2 } may be due to Bose-Einstein condensates with repulsive ( k > 0 ) or attractive ( k < 0 ) self-interaction , or have another origin .
In this paper , we consider positive indices n > 0 .
In that case , the polytropic component dominates in the early universe where the density is high .
For \alpha = 1 / 3 , n = 1 and k = -4 / ( 3 \rho _ { P } ) , we obtain a model of early universe describing the transition from a pre-radiation era to the radiation era .
The universe exists at any time in the past and there is no singularity .
However , for t < 0 , its size is less than the Planck length l _ { P } = 1.62 10 ^ { -35 } { m } .
In this model , the universe undergoes an inflationary expansion with the Planck density \rho _ { P } = 5.16 10 ^ { 99 } { g } / { m } ^ { 3 } that brings it to a size a _ { 1 } = 2.61 10 ^ { -6 } { m } at t _ { 1 } = 1.25 10 ^ { -42 } { s } ( about 20 Planck times t _ { P } ) .
For \alpha = 1 / 3 , n = 1 and k = 4 / ( 3 \rho _ { P } ) , we obtain a model of early universe with a new form of primordial singularity : The universe starts at t = 0 with an infinite density and a finite radius a = a _ { 1 } .
Actually , this universe becomes physical at a time t _ { i } = 8.32 10 ^ { -45 } { s } from which the velocity of sound is less than the speed of light .
When a \gg a _ { 1 } , the universe evolves like in the standard model .
We describe the transition from the pre-radiation era to the radiation era by analogy with a second order phase transition where the Planck constant \hbar plays the role of finite size effects ( the standard Big Bang theory is recovered for \hbar = 0 ) .